Real Roots Research Articles

Chebyshev subdivision and reduction methods for solving multivariable systems of equations

We present a new algorithm for finding isolated zeros of a system of real-valued functions in a bounded interval in Rn. It uses the Chebyshev proxy method combined with a mixture of subdivision, reduction methods, and elimination checks that leverage special properties of Chebyshev polynomials. We prove the method has quadratic convergence locally near simple zeros of the system. It also finds all nonsimple zeros, but convergence to those zeros is not guaranteed to be quadratic. We also analyze the arithmetic complexity and the numerical stability of the algorithm and provide numerical evidence in dimensions up to five that the method is both fast and accurate on a wide range of problems. Our tests show that the algorithm outperforms other standard methods on the problem of finding all real zeros in a bounded domain. Our Python implementation of the algorithm is publicly available at https://github.com/tylerjarvis/RootFinding.

$\mathfrak{sl}_{2}$ Triples Whose Nilpositive Elements Are in a Space Which Is Spanned by the Real Root Vectors in Rank 2 Symmetric Hyperbolic Kac–Moody Lie Algebras

In analogy with the theory of nilpotent orbits in finite-dimensional semisimple Lie algebras, it is known that the principal \mathfrak{sl}_{2} subalgebras can be constructed in hyperbolic Kac–Moody Lie algebras. We obtained a series of \mathfrak{sl}_{2} subalgebras in rank 2 symmetric hyperbolic Kac–Moody Lie algebras by extending the aforementioned construction. We present this result and also discuss \mathfrak{sl}_{2} modules obtained by the action of the \mathfrak{sl}_{2} subalgebras on the original Lie algebras.

A novel fluorescent probe for viscosity and polarity detection in real tobacco root cells and biological imaging.

The disruption of lipid droplet function is associated with the pathogenesis of various diseases. Clarifying the response behavior of lipid droplets to the microenvironment at the cellular level is of great significance. Plant lipids not only exist in phospholipids in cell membranes, but also in aromatic essential oils. Monitoring the level of lipid droplets in plant cells using fluorescent probes provides a simple method for screening lipid-rich varieties. We synthesized a polarity-viscosity responsive coumarin fluorescent probe, Cou-CN, which achieved sensitive detection of polarity and viscosity in dilute solution environments by constructing this simple probe with ICT and TICT properties and verifying it using Gaussian computational simulation. Cou-CN exhibited good lipid droplet illumination effects in HepG2 cells with a correlation coefficient of 0.92 compared to the commercial lipid droplet dye BODIPY. Additionally, co-staining the probe with the lipophilic commercial dye Nile Red in tobacco root stem seedling cells resulted in a high correlation coefficient of 0.9.

Design of HTS linear phase bandpass filter based on the inline topology

Abstract Inline topology filters have the advantages of a simple structure and fewer coupling coefficients. Based on the inline topology, this paper constructs a phase compensation topology of the filter, which can independently and controllably introduce two types of transmission zeros (TZs), and is convenient to realize the Chebyshev-like function response and compensate group delay. The finite frequency transmission zero is directly introduced through the source and load (S/L), and the real zero point is introduced and adjusted through the cross-coupling unit of the main path in this inline phase compensation topology. The two types of TZs can be introduced and regulated independently, so the side-band suppression and group delay compensation can be freely adjusted. A High-temperature superconducting (HTS) filter with inline is designed and processed with a typical structure based on application requirements. The filter introduces two pairs of symmetrical real TZs to improve sideband rejection and a pair of real TZs to control group delays to realize the linear phase. In addition, it has a small size and straightforward construction, and the results of the full-wave analysis method and the theoretical analysis approach match well with the measured results of the frequency response characteristics.

On random polynomials with an intermediate number of real roots

For each α ∈ ( 0 , 1 ) \alpha \in (0, 1) , we construct a bounded monotone deterministic sequence ( c k ) k ⩾ 0 (c_k)_{k \geqslant 0} of real numbers so that the number of real roots of the random polynomial f n ( z ) = ∑ k = 0 n c k ε k z k f_n(z) = \sum _{k=0}^n c_k \varepsilon _k z^k is n α + o ( 1 ) n^{\alpha + o(1)} with probability tending to one as the degree n n tends to infinity, where ( ε k ) (\varepsilon _k) is a sequence of i.i.d. (real) random variables of finite mean satisfying a mild anti-concentration assumption. In particular, this includes the case when ( ε k ) (\varepsilon _k) is a sequence of i.i.d. standard Gaussian or Rademacher random variables. This result confirms a conjecture of O. Nguyen from 2019. More generally, our main results also describe several statistical properties for the number of real roots of f n f_n , including the asymptotic behavior of the variance and a central limit theorem.

The Newman algorithm for constructing polynomials with restricted coefficients and many real roots

The Newman algorithm for constructing polynomials with restricted coefficients and many real roots

Point evaluation in Paley–Wiener spaces

We study the norm of point evaluation at the origin in the Paley–Wiener space PWp for 0 < p < ∞, i.e., we search for the smallest positive constant C, called Cp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr{C}}_{p}$$\\end{document}, such that the inequality |f(0)|p≤C∥f∥pp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\vert f(0)\\vert ^{p}\\leq C \\Vert f\\Vert _{p}^{p}$$\\end{document} holds for every f in PWp. We present evidence and prove several results supporting the following monotonicity conjecture: The function p↦Cp/p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p \\mapsto {\\mathscr{C}}_{p}/p$$\\end{document} is strictly decreasing on the half-line (0, ∞). Our main result implies that Cp<p/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr{C}}_{p} < p/2$$\\end{document} for 2 < p < ∞, and we verify numerically that Cp>p/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr{C}}_{p} > p/2$$\\end{document} for 1 ≤ p < 2. We also estimate the asymptotic behavior of Cp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr{C}}_{p}$$\\end{document} as p → ∞ and as p → 0+. Our approach is based on expressing Cp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr{C}}_{p}$$\\end{document} as the solution of an extremal problem. Extremal functions exist for all 0 < p < ∞; they are real entire functions with only real zeros, and the extremal functions are known to be unique for 1 ≤ p < ∞. Following work of Hörmander and Bernhardsson, we rely on certain orthogonality relations associated with the zeros of extremal functions, along with certain integral formulas representing respectively extremal functions and general functions at the origin. We also use precise numerical estimates for the largest eigenvalue of the Landau–Pollak–Slepian operator of time-frequency concentration. A number of qualitative and quantitative results on the distribution of the zeros of extremal functions are established. In the range 1 < p < ∞, the orthogonality relations associated with the zeros of the extremal function are linked to a de Branges space. We state a number of conjectures and further open problems pertaining to Cp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr{C}}_{p}$$\\end{document} and the extremal functions.

Rings and C*-algebras generated by commutators

We show that a unital ring is generated by its commutators as an ideal if and only if there exists a natural number N such that every element is a sum of N products of pairs of commutators. We show that one can take N≤2 for matrix rings, and that one may choose N≤3 for rings that contain a direct sum of matrix rings – this in particular applies to C*-algebras that are properly infinite or have real rank zero. For Jiang-Su-stable C*-algebras, we show that N≤6 can be arranged.For arbitrary rings, we show that every element in the commutator ideal admits a power that is a sum of products of commutators. Using that a C*-algebra cannot be a radical extension over a proper ideal, we deduce that a C*-algebra is generated by its commutators as a not necessarily closed ideal if and only if every element is a finite sum of products of pairs of commutators.

Strong versions of impulsive controllability and sampled observability

We give a simple proof of the (perhaps not so) well known fact that exponential polynomials of order k with real exponents have at most k−1 real zeros. We deduce several results that relate to impulsive controllability and sampled observability of finite-dimensional linear time-invariant dynamical systems. We prove that the initial state of a continuous-time linear time-invariant dynamical system of dimension n can be uniquely reconstructed from the sampled output regardless of its sampling time sequence of length n if and only if the system is observable and all the eigenvalues of the system matrix are real. This result thus characterizes sampled observability with arbitrary sampling times. Likewise, we prove that the system is controllable by means of n impulses regardless of when they occur if and only if the system is controllable and all the eigenvalues of the system matrix are real. We also show that, if the system is observable and all the eigenvalues of the system matrix are real, then there exists an initial state such that the times where the output crosses a prescribed threshold are prescribed m times with m<n.

Representing Matroids over the Reals is $\exists \mathbb R$-complete

A matroid $M$ is an ordered pair $(E,I)$, where $E$ is a finite set called the ground set and a collection $I\subset 2^{E}$ called the independent sets which satisfy the conditions: (i) $\emptyset \in I$, (ii) $I'\subset I \in I$ implies $I'\in I$, and (iii) $I_1,I_2 \in I$ and $|I_1| &lt; |I_2|$ implies that there is an $e\in I_2$ such that $I_1\cup \{e\} \in I$. The rank $rank(M)$ of a matroid $M$ is the maximum size of an independent set. We say that a matroid $M=(E,I)$ is representable over the reals if there is a map $\varphi \colon E \rightarrow \mathbb{R}^{rank(M)}$ such that $I\in I$ if and only if $\varphi(I)$ forms a linearly independent set. We study the problem of matroid realizability over the reals. Given a matroid $M$, we ask whether there is a set of points in the Euclidean space representing $M$. We show that matroid realizability is $\exists \mathbb R$-complete, already for matroids of rank 3. The complexity class $\exists \mathbb R$ can be defined as the family of algorithmic problems that is polynomial-time is equivalent to determining if a multivariate polynomial with integers coefficients has a real root. Our methods are similar to previous methods from the literature. Yet, the result itself was never pointed out and there is no proof readily available in the language of computer science.

Ring class fields and a result of Hasse

For squarefree d>1, let M denote the ring class field for the order Z[−3d] in F=Q(−3d). Hasse proved that 3 divides the class number of F if and only if there exists a cubic extension E of Q such that E and F have the same discriminant. Define the real cube roots v=(a+bd)1/3 and v′=(a−bd)1/3, where a+bd is the fundamental unit in Q(d). We prove that E can be taken as Q(v+v′) if and only if v∈M. As byproducts of the proof, we give explicit congruences for a and b which hold if and only if v∈M, and we also show that the norm of the relative discriminant of F(v)/F lies in {1,36} or {38,318} according as v∈M or v∉M. We then prove that v is always in the ring class field for the order Z[−27d] in F. Some of the results above are extended for subsets of Q(d) properly containing the fundamental units a+bd.

Estimating Bethe roots with VQE

Abstract Bethe equations, whose solutions determine exact eigenvalues and eigenstates of corresponding integrable Hamiltonians, are generally hard to solve. We implement a Variational Quantum Eigensolver approach to estimating Bethe roots of the spin-1/2 XXZ quantum spin chain, by using Bethe states as trial states, and treating Bethe roots as variational parameters. In numerical simulations of systems of size up to 6, we obtain estimates for Bethe roots corresponding to both ground states and excited states with up to 5 down-spins, for both the closed and open XXZ chains. This approach is not limited to real Bethe roots.

Transient stability of grid following inverters: The impacts of damping and fault ride‐through

AbstractThe interaction of grid following inverters with a weak grid raises risks of transient instability. The effects of damping and fault‐ride through (FRT) make the transient dynamics more complicated. To investigate the impact of FRT on the transient stability of converters, this paper first employs the switch system theory to construct a mathematical model for the synchronous mechanism of inverters. Secondly, the method of semi‐algebraic system real root classification is comprehensively used to analyse the existence of equilibrium points (EPs) and provide necessary and sufficient conditions for their existence. Finally, based on the theory of differential geometry, a method for computing the stability domain of converters is proposed. Compared with traditional methods, the results obtained from this approach are precise and non‐conservative. The study in this paper indicates that the equivalent damping effect from the proportional control in a phase‐locked loop significantly enlarges the stability region, whereas the active current injection shrinks the stability region. The reactive current droop characteristic also has impacts on the stability region, especially with active current injection. The switchover between the droop mode and constant current mode requires extra precaution in the calculation of the EPs and critical fault clearing time.

Structure and arithmetic of multivariate Ore extensions

We give the basic structure of the multivariable Ore extensions S = A [ t ¯ ; σ , δ ¯ ] introduced in the work of Martínez-Peñas and Kschischang. The Pseudo multilinear transformations (PMT’s) are introduced and correspond to modules over S. These maps are strongly connected to the evaluation of polynomials in S. A general product formula is obtained. PMT’s help to put some structure on the set of roots of a polynomial f ( t ) ∈ S .

Zero-g wind tunnel experiments on planetesimal erosion

ABSTRACT Wind erosion thresholds for planetesimals in protoplanetary discs depend on particle size, cohesion, ambient gas pressure, and gravity. In the experiments reported here, we used a low pressure shear wind tunnel for the first time under real zero gravity. That is, we carried out the experiments in the drop tower in Bremen at $10^{-5} \text{g}$. In concert with earlier work, we confirm the applicability of a single semi-empirical equation to describe the threshold shear stress for wind erosion. It can be applied to calculations of unstable orbits of planetesimals and the dissolution of planetesimals passing through evolving planetary atmospheres.

Complex Roots-Finding Method of Nonlinear Equations

Various researchers developed numerical methods to solve nonlinear equations. But still, there is a need to improve these methods to get accurate results. Some techniques exist where convergence is not assured. These methods increase the problem's cost using a second/higher-order derivative. So, cost efficiency is also considered an issue in many fast-converging ways. This paper proposes a new numerical method to find complex and real roots of nonlinear equations. By taking an initial guess w0 ∈ C, the new approach contributes to a series of iterations converging to a real or complex solution. Moreover, the method evaluates complex roots even if a real number is taken as an initial guess. This innovation presents that the proposed method achieves second-order convergence. The number of iterations is also determined to check the performance of the new iterative scheme. Using Python 3.10.9 package, the authors have tested the method's efficacy on several numerical problems, presented the results in the tables, and illustrated them with the help of graphs.

Central Limit Theorem for the number of real roots of random orthogonal polynomials

Central Limit Theorem for the number of real roots of random orthogonal polynomials

Fast evaluation and root finding for polynomials with floating-point coefficients

Evaluating or finding the roots of a polynomial f(z)=f0+⋯+fdzd with floating-point number coefficients is a ubiquitous problem. By using a piecewise approximation of f obtained with a careful use of the Newton polygon of f, we improve state-of-the-art upper bounds on the number of operations to evaluate and find the roots of a polynomial. In particular, if the coefficients of f are given with m significant bits, we provide for the first time an algorithm that finds all the roots of f with a relative condition number lower than 2m, using a number of bit operations quasi-linear in the bit-size of the floating-point representation of f. Notably, our new approach handles efficiently polynomials with coefficients ranging from 2−d to 2d, both in theory and in practice.

Algebraic winding numbers

In this paper, we propose a new algebraic winding number and prove that it computes the number of complex roots of a polynomial in a rectangle, including roots on edges or vertices with appropriate counting. The definition makes sense for the algebraic closure C=R[i] of a real closed field R, and the root counting result also holds in this case. We study in detail the properties of the algebraic winding number defined in [3] with respect to complex root counting in rectangles. We extend both winding numbers to rational functions, obtaining then algebraic versions of the argument principle for rectangles.

Unimodal sequences: From Isaac Newton to June Huh

In this paper, we present a simple proof of a famous result of Newton on unimodal sequences associated to polynomials with real roots. We then give a brief exposition of the recent work of Huh on such sequences arising from combinatorial geometries.